A spherical thermocouple junction of diameter 0.706 mm is

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A spherical thermocouple junction of diameter 0.706 mm is to be used for the measurement of temperature of a gas stream. The convective heat transfer coefficient on bead surface is 400 W/m²K, Thermophysical properties of thermocouple material are k = 20 W/mK, c = 400 J/kgK and ρ = 8500 kg/m³. If the thermocouple initially at 30°C is placed in a hot stream of 300°C, the time taken by the bead to reach 298°C, is

1. 2.35 s
2. 4.9 s
3. 14.7 s
4. 29.4 s

Correct Option: B

Given h = 400 W/m²K, k = 20 W/mK
c = 400 J/kg.K, ρ = 8500 kg/m³ T(t) = 298 °C, T_i = 30 °C, T_∞ = 300 °C

L =	V	=	(π / 6)D³	=	D
	A		πD²		6

=	1	× 0.706 × 10^-3 = 1.76 × 10^-4 m
	6

Biot Number =	hL
	k

=	400 × 1.176 × 10^-4	= 0.0023
	20

Since B_i < 0.1, lumped system analysis can be used

	T(t)	= e	-hAT
	T_i - T_∞		ρcv

	298 - 300	= e	-hAT
	30 - 300		ρcV

	-2	= e	-hAT
	270		ρcV

ln		270		=	hAT
		2			ρcV

⇒ T =	ρcV	ln		270		=	ρc		V		ln 135
	cA			2			h		A

=	8500 × 400 × 1.176 × 10^-4 × 4.90	≈ 4.9 sec.
	400